polynomial and remainders (1 Viewer)

[failingTheHsc]

anyone help me with this:

when the polynomial P(x) is divided by (x+1)(x-4), the quotient is Q(x) and the remainder is R(x).
i) Why is the most general form of R(x) given be R(x) = ax + b
ii) Given that P(4) = - 5, show that R(4) = -5
iii) Further, when P(x) is divided by (x+1), the remainder is 5. Find R(x)

 

[kimmeh]

hey this is in a past paper
i did this last night

a)
since P(x), when expanded, has a degree of 2, when divided, the degree of R(x) must be smaller degree than 2. thus R(x) must be in the form ax + b


b)
P(x) = (x+1) (x-4) . Q(x) + R(x)
.: P(4) = 5 x 0 x Q(4) + R(4)
.: P(4) = R(4)
and since P(4) = -5
.: R(4) = -5

c) R(x) = ax = b
R(4) = -5 --> 4a + 5 = -5 [1]
R(-1) = 5 --> -a + b = 5 [2]

[1] - [2]
.: 5a = -10
.: a = -2

and b = 3

.: R(x) = -2x + 3

 

[failingTheHsc]

thanx i *almost* understand it.

how do u know it has a degree 2?

 

[freaking_out]

Originally posted by failingTheHsc
thanx i *almost* understand it.

how do u know it has a degree 2?

well just expand out P(x) and u'll get a quadratic-which is degree 2.